Find Height of a Flagpole without Approaching the Pole

Date: 04/13/2006 at 10:12:02
From: Patrick
Subject: find height of flagpole, with no distance to pole

I am outside a fence at a right angle to a pole of unknown distance or 
height.  I can measure along the fence and take the angles from 
ground level to the top of the pole from two points, exactly level
with the base of the pole and 80 m apart.

Points A, B, and C are exactly on level ground, A to B is 80 meters,
the angle at B is a right angle.  On C is a vertical pole with the top
of the pole = D.  Angle CBD = 32 degrees and angle CAD = 24 degrees.

I am 56 yrs old and when I learned math there were no calculators.  I
was given this wee puzzle, found your brilliant site, and learned
something about trig.  Alas when this puzzle combined three dimensions 
with algebra and trig it was sink or swim--time I learned to swim!

I can imagine two equations being equal to each other.  Tan(32) = CD
over BC and Tan(24) = CD over AC with CD being common so Tan(32) x BC
= Tan(24) x AC.  From this I know that there would be a ratio between
distances AD and BD...I believe it is workable, maybe over my head but 
I'm interested in the answer.  Thank you.

Date: 04/13/2006 at 11:35:15
From: Doctor Douglas
Subject: Re: find height of flagpole, with no distance to pole

Hi Patrick.

Good work so far!  From your description I gather that angle CBA is a
right angle, so that on the line that passes through A and B, the
point B is in fact the closest point to the flagpole:

  

Let h be the height of the flagpole CD.  Also, let y denote the
distance between B and C and Y denote the distance between A and C.

The elevation angles give

  h = y tan(32) 
  h = Y tan(24)

The additional information that gives the relationship between the
distances is the Pythagorean Theorem.  Although I apply this to
triangle ABC below, you can also apply it to triangle ABD, since that
is a right triangle also.  Because triangle ABC is a right triangle,
we have Y^2 = y^2 + 80^2, and our two equations for the height h can
be set equal to each other and rewritten in terms of y:

  y tan(32) = sqrt(y^2 + 6400) tan(24)

and we can square this and solve for y:

  y^2 tan^2(32) = y^2 tan^2(24) + 6400 tan^2(24)

  y = sqrt{[6400 tan^2(24 deg)]/[tan^2(32 deg) - tan^2(24 deg)]}
    = sqrt(6600)
    = 81.24 meters

And we can now calculate the height of the flagpole as 

  h = y*tan(32 deg) = (81.24 m)*(.6249) = 50.76 meters.  

As a check, we also compute Y = sqrt(81.24^2 + 6400) = 114.02 meters, 
and (114.02 meter)*tan(24 deg) = 114.02*(.4452) = 50.76 meters, in
agreement.  So everything is consistent and we can be confident that
our work is correct.

For another approach to measuring the height of a pole without access
to the base point C, see the following web page in our archives:

  Determine Flagpole Height without Access to the Pole
    http://mathforum.org/library/drmath/view/65178.html 

In that answer we try to keep the geometry confined to a 
two-dimensional plane, so that it is easier to visualize and to draw
on paper.  The price that we have to pay is that we have to apply
trigonometry to a triangle that is not a right triangle.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/16/2006 at 09:15:15
From: Patrick
Subject: Thank you (find height of flagpole, with no distance to pole)

Very kind of you to afford your knowledge and time to satisfy my
curiosity.  Bless you all, I hope to be able to pass on what I have
learned from you.  Thanks, Patrick